A113300 - OEIS

%I #35 Sep 08 2022 08:45:23

%S 0,1,3,10,34,115,389,1316,4452,15061,50951,172366,583110,1972647,

%T 6673417,22576008,76374088,258371689,874065163,2956941266,10003260650,

%U 33840788379,114482567053,387291750188,1310198605996,4432370135229,14994600761871,50726371026838

%N Sum of even-indexed terms of tribonacci numbers.

%C A000073 is the tribonacci numbers. A099463 is the bisection of the tribonacci numbers.

%C Primes in this sequence include a(2) = 3, a(6) = 389, a(9) = 15061, a(10) = 50951. a(n) in this sequence is semiprime for n = 3, 4, 5, 11, 14, ...

%C Partial sums of A099463. a(n+1) gives row sums of Riordan array (1/(1-x)^2,(1+x)^2/(1-x)^2)). Congruent to 0,1,1,0,0,1,1,0,0,... modulo 2. - _Paul Barry_, Feb 07 2006

%H G. C. Greubel, <a href="/A113300/b113300.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,1).

%F a(n) = Sum_{i=0..n} A000073(2*n).

%F a(n) = Sum_{i=0..n} A099463(n).

%F a(n) + A113301(n) = A008937(n).

%F From _Paul Barry_, Feb 07 2006: (Start)

%F G.f.: x/(1 - 3*x - x^2 - x^3).

%F a(n) = 3*a(n-1) + a(n-2) + a(n-3). (End)

%t Accumulate[Take[LinearRecurrence[{1,1,1},{0,0,1},60],{1,-1,2}]] (* _Harvey P. Dale_, Nov 06 2011 *)

%t LinearRecurrence[{3,1,1},{0,1,3},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)

%t a[ n_] := Sum[ SeriesCoefficient[ SeriesCoefficient[ x / (1 - x - y - x y) , {x, 0, n - k}]^2 , {y, 0, k}], {k, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)

%o (Magma) I:=[0,1,3]; [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..61]]; // _G. C. Greubel_, Nov 19 2021

%o (Sage)

%o @CachedFunction

%o def T(n): # T(n) = A000073(n)

%o     if (n<2): return 0

%o     elif (n==2): return 1

%o     else: return T(n-1) +T(n-2) +T(n-3)

%o def a(n): return sum( T(2*j) for j in (0..n) )

%o [a(n) for n in (0..60)] # _G. C. Greubel_, Nov 19 2021

%Y Cf. A000073, A008937, A099463, A113301.

%K easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Oct 24 2005